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User blog:Holomanga/Resourcespace
This blog post is available in .pdf form at . One way of thinking about civilisational growth is to imagine them exploring a real space out to some distance vt , and embedded in this real space are resources (typically stars). This is a useful framing for polynomial growth, in which this is actually what occurs, but is somewhat unintuitive for exponential and hyperbolic growth, in which most growth occurs in one physical location. Resourcespace is another way of thinking about the expansion of a civilisation. In resourcespace, the distance between two points is measured in units of resources, rather than in length. A civilisation begins at the origin - equivalent to having no resources - and then explores resourcespace out to a distance R equal to the amount of resources it has access to. If resource generators are embedded in this space, then qualitatively this represents resources being expended to build resource generators (hunters, farms, factories), which themselves produce more resources. In an n -dimensional resourcespace with a constant resource generator density \rho , the rate of resource generation is hence \dot R = \rho V® , where V is the volume enclosed by a sphere of radius R in resourcespace. For a one-dimensional resourcespace, V® = 2R , so growth is exponential. This intuitively matches our model of the causes of exponential growth - each unit of resources being a resource generator that is used to generate some more resources. But, most interestingly, if n < 1 , we have polynomial growth, and if n > 1 , we have hyperbolic growth. These three apparently qualitatively different forms of growth in fact depend on the (fractal) dimensionality of the resourcespace the civilisation is currently exploring! What does the geometry of a typical resourcespace look like? My standard model of civilisation growth is a series of exponential growth modes (the forager, farmer and industrial eras), followed by a hyperbolic growth mode (the computational era), followed by a series of polynomial growth modes (the galactic and cosmic eras). This behaviour suggests a resourcespace geometry qualitatively similar to Figure 1. The three labelled features are the primitive line, the intelligence bulge, and the long tail. A civilisation, initially beginning at the origin, expands along the primitive line, contructing resource collectors, while also slowly exploring the intelligence bulge which is much less lucrative but contains technologies related to intelligence augmentation, such as computers and mind uploading (the two axes can be thought of roughly as one corresponding to having more resources, and one corresponding to thinking faster due to having those resources). Eventually, the \propto R^2 growth of resource generation rate in the intelligence bulge dominates over the \propto R growth of resource generation in the primitive line, and the civilisation transitions to the singularitarian growth regime and rapidly occupies the entire intelligence bulge. At that point, the only growth left is in the long tail, which represents distant resources that can only be obtained by physically reaching them, and which is tapered such that it has fractal dimension \frac{1}{3} . ρ Values Along the primitive line, the growth rate has been well-studied, which means that it is possible to calculate the density of resource generators. Along this line, \dot R = 2\rho R + O(R^2) , with the R^2 term (and possibly higher-order terms) from intelligence augmentation that can be neglected for primitive civilisations. This means that R = R_0 e^{2\rho t} . The characteristic timescale of this exponential growth is \tau = \frac{1}{2 \rho} , so the density of resource generators is \rho = \frac{1}{2 \tau} . These are shown for the three major exponential growth regimes in Table 1. Roughly, the density of resource generators in a small region \delta R represents how many times that region can replicate itself per unit time, and the number of resource generators is the actual rate at which it produces resources. We can also put a bound on the density of the intelligence bulge. Because the R^2 term is negligible today, we know that \rho_\mathrm{e} R_t \gg \rho_\mathrm{h} R_t^2 where \rho_\mathrm{e} is the current density of resource generators at the frontier of the primitive line for modern humanity, \rho_\mathrm{h} is the density of resource generators in the intelligence bulge, and R_t is the current amount of resources available to human civilisation. Rearranging this gives \rho_\mathrm{h} \ll \frac{\rho_\mathrm{e}}{R_t} . For modern humanity, {\rho_\mathrm{e} \approx 1.6 \times 10^{-2}\, \text{y}^{\text{-1}}} , and {R_t \approx 21\times 10^6 \, \text{MW}} , so {\rho_\mathrm{h} \ll 10^{-9} \, \text{y}^{\text{-1}} \, \text{MW}^{\text{-1}}} . Category:Blog posts